378 IEEE TRANSACTIONS ON ANTENNAS AND PROP.4GATION, VOL. AP-19, NO. 3, a Y 1971

Multiple Scattering of EM Waves by Spheres Part I-Multipole Expansion and

Ray-Optical Solutions

Abstract-Solution to the multiple scattering of electromag- netic (EM) waves by two arbitrary spheres has been pursued first by the multipole expansion method. Previous attempts at numerical solution have been thwarted by the complexity of the translational addition theorem. A new recursion relation is derived which reduces the computation effort by several orders of mag- nitude so that a quantitative analysis for spheres as large as lox in radius at a spacing as small as two spheres in contact becomes feasible. Simplification and approximation for various cases are also given. With the availability of exact solution, the usefulness of various approximate solutions can be determined quantitatively. For high frequencies, the ray-optical solution is given for two conducting spheres. In addition to the geometric and creeping wave rays pertaining to each sphere alone, there are rays that undergo multiple reflections, multiple creeps, and combinations of both, called the hybrid rays. Numerical results show that the ray-optical solution can be accurate for spheres as small as x/4 in radius is some cases. Despite some shortcomings, this ap- proach provides much physical insight into the multiple scattering phenomena.

~.

INTRODUCTION

T HE SIMPLEST realistic problem of multiple scat- tering by finite bodies appears to be that. by two

spheres. Many works [l>[9] on this subject can be found in the literature, but most either deal with’ general formulation or are confined to specific cases, and prac- tically none give numerical result,s. Even the limited amount. of experimental works by Mevel [a] and Angelakos and Kumagai [5] are in need of independent. verificat.ion and extension.

It is not the intent of this paper to include a general survey of all the works on this subject, for which the readers are referred to the excellent reviews by Twersky and Burke [lo], [ll] and also [E] . Here citations Kill be made only to closely related works.

in part by NSF under Grant, GK4161 and in part by t.he Bendix Xanuscript received August 3, 1970. This work was supported

Missile System Division under a Research Grant. This paper is based in part on a dissertation submit.ted by J. H. Bruning in part.ia1 fuliillment of t.he requirements for the Ph.D. degree.

J. H. Bruning was with the Antenna Laboratory, University of

rat.ories, Murray Hill, N. J. 07971. Illinois, Urbana, Ill. 61801. He is now with Bell Telephone Labo-

Urbana, Ill. 61801. Y . T. Lo is with t.he Antenna Laboratory, University of Illinois,

I n 1935 Trinks [l] considered scattering by t,nro iden- t.ical spheres of radii much smaller than wa,velength, which was later extended to small unequal spheres and arbitrary a.ngle of incidence by Germogenova [4]. More recently Liang and Lo [7] and Crane [9] reformulated t,he prob- lem using a newly derived translational addition theorem by Stein [13] and Cruzan [14] whereas Tmersky [SI considered a. more general problem with many scatterers, using dyadic Green’s function approach. He also obtained a,pproximate solutions under various conditions.

More recently Levine and Olaofe [lj] extended Trinks’ work to arbitrary orientation of t.wo small part.icles and also considered the effect of t.he electric quadrupole. Even with the help of several previous theoretical works and the availability of modern high speed computers, Liang and Lo [7] found that their numerical evaluat.ion had to be limited to spheres of radii less than 3h/4 and wide spacings, due to t,he complexity of the addition theorem. This suggests that the numerical aspect. of t.he problem is by no means trivial.

In this paper t.he additiona.1 theorem as applied to the present problem is reexamined from a numerical point. of view. -4n important recursion relation is introduced which permits a routine calculation of t,he t,ranslation coefficient,s without. resort.ing to the t,ime-consuming computation of Rigner’s 3-3’ symbols [7]. In doing so, the comput,ing effort can be reduced by several orders of magnit,ude. As a result, quant.it,ative analyses for spheres as large as 1OX in radius, of arbitrary materials, even in contact be- come feasible. Closed form approximate solut.ions under various conditions are also given for the purpose of deter- mining their validity by comparing them with exact. ones numericaIly.

Like all other scat.tering problems, a multipole expan- sion solution loses its effectiveness a t high frequencies. Therefore, in that. case, a ray-optical solution is very de- sirable. Furthermore, this type of solution offers much physical insight. into the complex multiple scattering mechanism. The ray-optical solution is based on the classical geometric optics and creeping wave t.heor;. With t.he except,ion of endfire incidence, t.he numerical results are in excellent. agreement with those obtained by mu1t)i- pole expansion for conducting spheres as small as h/4

BRTJNING AND LO: MULTIPLE SCATTERLKG OF RAYES BY SPHERES, I 379

in some cases. It is grat,ifying to see that there exists z

such a large overhpping region (X/&lOX) between two types of solutions which gives us ample latitude for cross- checks of the results. The confidence on t,hese solutions is further strengt,hened by observing amazingly close agreement, with the experiment.al results which are dis- 1

cussed in [a;].

STATEMENT O F PROBLEM AiiD MULTIPOLE EXPANSIOK d SOLLTIOW 6

Consider two spheres A and B of arbitrary materials and radii a and b, respectively. Without loss of generality, x !

their centers 0 and Of , separa.t.ed by a dist,ance d, can be assumed to lie on the z a.xis, as shown in Fig. 1. Any point in space ca.n be represented by (r,e,4) or (r’,0’,4) with respect, to t.he coordinate system with origins a t 0 or a.t. 0’, which is related to t,he former through a t,ransla- tion d along the z axis. i n t.he follo\+<ng, we sha.11 adopt the convent.ion that all unprimed quantities are referred to t.he 0 system whereas all primed are referred t.0 the 0‘ system. Let. there be an incident plane wave of unit st,rengt,h a.nd chara.cterized by a wave vector k, an inci- dent, angle a witith respect to t.he z axis, a.nd a polalization angle y between E and t.he projection of 00’ on t*he incident wavefront. as shown in Fig. 1. Then, following Strat,ton’s notations [lG] for vector spherical wave functions, t,he

Fig. 1. Geometry of two-sphere problem.

NmnfJ’),Mmn(j) are vector spherical wave functions

total electric field expanded with respect to 0 [7]-[9], [l2] can be written as,‘ for T 2 a,

Mmn(J’) = z,(j) ( k r ) (exp im4)

Y

where p (m.,n) and y (m,n) are mult.ipole c0efficient.s of -[ ag~ ,m(cos e) 6 + - p,m(cOs e) 4 t.he incident plane wave

iV1

sm 8

p (m,n) = in+‘ 2n + 1 (n - m ) !

n(n + 1) (n + m ) !

where z,(j) is the appropriate kind of spherical Bessel funct.ions j,, f2n, hn(l), and hnc2), for j = 1, 2, 3, and 4, respectively; AE,AH,BB,BH are “mult.ipole coefficients” of

-C7rmn(a) cos Y + irmn(a) sin r] (2) E and H waves scat.tered, respectively, by spheres A and B in t,he presence of each other; a.nd AmPmn,Bmymn are “t,ransla,t,ion coefficients” of the vector spherical wave functions from 0‘ to 0. (See the Appendix for det,ails).

There is a similar expression for ET’ wiiith respect to the 0‘ system, for which the muhipole coefficient of t.he inci- dent. plane wave p‘( m,n) and p‘(7r1,n) differ from p (m,n) and p(nz,n) by a factor exp ikd cos a, a.nd A‘,,””

Xmn(a) = daPnm(COSa), Trnn(a ) = 7 Pnrn(cOSa) (4) and Bfmvmn differ from Ampmn and Bmvmn by (-1)“+, and ( - l)n+v+l, respectively. The total magnetic field HT is obt.ained by intercha,nging N,, and M,, in (1) and

2n + 1 (12 - m ) ! n(n + I) (n. + m ) !

q(nz.,n) = in+’

- [ ~ , ~ ( a ) cos Y + irmn(a) sin Y] (3)

m sin a

Time variation exp ( - i d ) is assumed. a, = a/aa. mukiplying the result. by - i k /wp .

. - >

380 IEEE TRLYSACTIONS ON ANTENNAS AND PROPAGATION, MAY 1971

By applying the appropriate boundary conditions and vantage. Thus, for the t,wo principal poIarizatiom using the orthogonality properties of the Legendre func- tions, we arrive at four sets of coupled, linear, simultaneous equations in the unknown coefficients:

The coefficients v,(kl,ka), u,(kl,ka), vn(k2,kb), and u,(k?,kb) are, respectively, the classical elect.ric and mag- netic mukipole coefficients of the external field of spheres A and B in isolation. k , kl, and h.2, are the xmvenumbers in the surrounding medium, sphere A , and sphere B, respectively.

I t should be emphasized that, the syst.em (7) is valid for determination of t,he mukipole coefficients for the scattering by any pair of spheres, equal or unequal in size, of the same or different. ma.teria1 as long a.s the appropriate single sphere coefficients un and L',, for each sphere a.re known.2

It. is clear from (7) that there is no coupling among the azimuthal modes (i.e., modes 1vit.h different index ~ n ) ; hence, this system of equations may be solved independ- ent.ly for ea.ch nz where -n 5 m 5 n, and represents in general, 2n + 1 sets of equations. As shov-n later, there

: are several important special cases in which the form of (7) can be simplified considerably.

SCATTERED FAR FIELD .4hr~ VARIOUS APPROXINATE SOLUTIONS

Of pract,ical interest' is the scat.tered field in the far zone of the ensemble of two spheres which may, neverthe- less, be in the near field of each other. This is obt.ained from the asymptotic forms of t.he vector spherical wave functions for r , T' >> d. By observing the symmetry in- herent. in the c0efficient.s and t,he wave functions, all previous field expressions can be t.ransfornled into series involving only nonnegative values of the index m if the incident field is decomposed inta it,s horizonta.1 a.nd ver- tical components which yields a grea,t comput.ationa1 a.d-

posed of several concent.ric layers of different materials [13] or * The precise form for u.,, and vn is also known for spheres com-

concentric layers of nonuniform dielectric constant. [24].

e (tn,n) = AE: (?~t,n) + B ~ ( m , n ) exp ( - ikd cos a)

h (,tn.,n) = AH(m,n) + BH (nz,n) exp ( -iM cos a). (9)

Note that e (m,n) and h (m,n) are also dependent on y by virtue of (a), (3), and (7).

There are several insta.nces in which t.he analysk may be simplified further by imposing certain additional re- strict.ions.

1) The fmt involves the case of axial symmetry, i.e., when the propa.gation direction of the incident field coin- cides with the axis of the two spheres (endfire incidence a = 0). As a consequence, t.he coefficients ( 2 ) and (3) of the incident field become

p (n2,n) = q ( n w )

where 6 m , , is t.he Kronecker delta.. This means that the system (7) need be solved only for ? n = 1, where t.he coefficients and Blvln assume a pa,rticularly simple form [12].

2) Another simplification involves identical spheres at. broadside incidence ( a = b, a = r / 2 ) where it may be shown tha.t the coefficients of the spheres bear the simple rela,tion

B ~ ( v L , ~ ) = F ( - 1 ) n + r n - 4 ~ ( ~ ~ , n )

BH(.nt,n) = f ( - 1) , + ' A H ( n 2 , n ) . (11)

The upper and lower signs refer, respectively, to the inci- dent, polarizations y = 0, ~ j 2 . With this simplification, (7) reduces to two coupled sets of equat.ions [12].

3) The Ra.yleigh approximation gives rise to a partic- ularly simple form, since this situation is characterized by

BRUKIXG AND LO: MULTIPLE SCATTERING O F R A V E S BY SPHERES, I

the assumption that. ka. and kb are so small that on ly the terms for n = 1 contribute, the ot.hers being t.a,ken as zero. This case has previously been considered by several authors using Trinks’ formulation [l], [2], [4], [SI. The explicit low-order translation coefficients are simply in- serted into (7), which is then solved algebraically [la].

4) Lastly, a very useful approximation is obtained for the case where ea.ch sphere is situated in each others’ far field. This is satisfied when d / z > 0 (kz) , where z is t,he larger of a. and b. In t.his case, it- can be shown t.hat the addition theorem takes a very simple form [12]. From t.his, we find tha.t. the system may be uncoupled and solved analytically. This is accomplished by successive subst.it.u- tion and not.ing t,hat in the process me are genera.t.ing geomet,ric series which can be summed (after a. consider- able amount, of bookkeeping) in closed form. For illustra- tion, consider the vertical polarizat.ion ( y = n/2) , and smttering in the plane r#J = n. Wit.h t,he identificat>ions

s,A = @SQb(d,n - ajo) + S+a(d,a,g)S@b(d~n;g)

1 - S+~(d,7r,7r)Xpyd,*,7r)

and

m

[ U n ( k d x l n ( e ) + u,(kz)nn(e)] cos 4

(13)

n=l

we arrive at [12]

E,A+B = S + a ( T , e + a , ~ ) + S,b(r,e + a , ~ ) exp is

+ sOBSOa(~ ,n - e,o) + s+AS+b(r,O,n) exp is

(14)

where 6 = kd(cos a - cos e). Inspection of (12)-( 14) re- minds us that we ha.ve a result composed only of single sphere watering a.mplitudes. Furthermore, this result. was obtained only by using the far field form of t.he addit.ion theorem.

The scattering amplitudes s+* and sgB have an enlighten- ing interpretation with the aid of Fig. 2. sQA, for example, is the st.rength of a. field incident on B from A . Its ampli- tude is composed of the first-order field scattered by B toward A : @SQb(d,n - a,O) and a field scattered by A to- ward B and then backscat.t.ered by B : S+a(d,a,n)Sgb(d,n,n). The term [l - SQa(d,n,n)SQb(d,n,n)]-’ multiplying tlhe aforementioned a,mplitudes is the effect of all higher order “bounces” of these two amplit.udes, which we note is t,he sum of a geometric series in powers of the term Spu(d,n,~)S+b(d,a,n); its convergence is assured by the

. 0 -

\

Fig. 2. InterpreCation of multiply scattered fields in (12) and (14).

initial assumption of each sphere being in t.he others’ far field. For two identical spheres a.t. broadside incidence, (14) becomes ( y = n / 2 )

(15)

It is interesting to note tha.t t.his approximat,ion yields surprisingly good results even for t,wo spheres in contact.. This is shown in Fig. 3, where the normalized radar cross sections (RCSs) for various sizes of conduct.ing spheres are plott.ed, the solid curves being obt,ained from the exact solut.ion.

RAY-OPTICAL SOLUTION

Levy and Keller [17] elegantly extended Franz’s creep- ing wave theory [18] for the sphere to sca.t.tering by a.n arbitra.ry smooth convex body. Ray paths associated with the creeping waves obey Fermat,’s principle and hence, lie along geodesics of the surface. While their approach to the elect.romagnetic (EM) problem makes use of t.wo scalar acoust.ic problems, here we are concerned specs- cally with the behavior of t.he vector problem of t,he sphere. Senior and Goodrich [19] expressed the single sphere scattered field in a form which makes ident.%cation of the a.ppropriate diffraction a.nd attenuation coefficients an easy matter.

The rays which contribut,e to t.he scattered field of the t.wo sphere ensemble fall int>o three categories: 1) reflect,ed rays arising from direct and mult.iple reflect.ions, 2) creep- ing wave rays bound to a single body, and 3) “hybrid raysJJ [E], [20]. The rays fa.lling into the last category

-. -

382 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, MAY 1971

OT Field calculabions employing the geometric optics ap-

0: e , - proximation are carried out, using Snell's Law and the conservation of energy. If E ( A ) is the value of the incident field at a point. A , then the reflected field a t some point a j k o i k b i 1 0 <,=-

0 $:I -. - 3 6376 P, a distance S from A , is given by

I f s ' o 2 j I . P I P 2 exp ikX (16)

where p1 and p2 are the principal radii of curvature of the wavefront reflected from A. The bracketed term is desig- nated the divergence factor A. If more than one reflection is involved, t,he preceding procedure is repeated; the field a t some point P after N reflections will then assume t.he

E ( P ) = - E ( A ) [(, + S) (pz + S ) 1 \- i i

9

1

8:m Lljm : sjm : , ~ m : 1b.m : A m ' A.m ' zS.m ' d.m ' form

8..

'. For simp1icit.y we will consider t.he scatt.ered field at 5..

k o = k b = 2 . 0 E ( P ) = ( - l ) N E ( A ) A, exp ikS,~. kd

hr

~ = l o o s l T O 2

.. '. -1

N ' s 8:,f0!_26 7y \ t.he point in t.he plane of incidence as shown in Fig. 4, where

only t,he member R2 of the set (Rj} is drawn. R2 consists of the subrays Sos - SE - SZ2. In genera.1, a member of

t.0 denote the length of the ray between reflection points i a.nd i + 1. The angle bet.ween ray Sij and the normal a t

4.. '. ' (R,} comprises Soj - SU - - - Sjj where Saj is used

a>:m : A, : I A . ~ : li.m : d m ' i m ' 26.m ' A.m ' 4.m ' t.he ith reflect.ion point is denot,ed by qi,. Then, for t4he kd assumed incident plane wave, t,he far field due to one

8.. k o = k b = 4 1 9 reflection R1 from sphere A is - = ua 06386 702 U

0 R1 = - - exp ik ( R + r - 2u COS 1/11) (17) 2r

where

7i'- e - Q!

2 r = S11 + a sin 1/11, 711 =

D

k:m ' 1b.m 1i.m ' 1i.m ' &.m ' S.m ' i . m ' 3v.m ' i . m ' k d

Fig. 3. RCS of t.wo equal metallic spheres at broadside incidence for X-a = 1.0, 2.0, and 4.19 using exact solut.ion (-----) and

. asymptotic form (15) for large separation (--.--.).

are t.hose which involve any combination of rays of the first two types. For purposes of further classification, rays of the &st type which undergo j reflections are denot.ed by Rj; those rays of the second type which creep over a length equal to or less than halfway around the body are denoted by C-; C+ describes the case of a la.rger length. Fina.lly, the third case may be represented by any com- bination of the preceding symbols with its obvious impli- cat'ion. For example, a ray which creeps part, way around one sphere, most. of the way around the other, and reflect.s five times between t.he two before reaching the observer may be identified by the symbol C-C+R5. A particular geometry of two spheres could support any number of configura6ions of rays; however, generally only a few will be significant..

6 is the polar angle of observation point P, and R is the distance from t,he source to origin 0. We have a similar expression for sphere B.

For two reflections, one must first. determine the t,hree unknowns ~ I Z , vn, and S, [E]. Once these parameters are determined, the divergence factors At may be calcu- lated. The result analogous to (17) for two reflections is given by [E]

-exp ikCR - a cos 1/12

+ SB + r - b cos 1/22 - d COS e] (18)

where r = SZ2 + b cos 771.2 + d cos 6 and

ll = a. + SE cos 1/12, L = S12 + a cos w l2 = b + 812 COS 1/22, L = SE $. b COS m .

There is, of course, a companion ray which reflects off of sphere B first and then A before reaching the observer. This may be calculated from t,he preceding by making a

BRUPI'ING AND LO: MULTIPLE SCATTERING OF W.4VES BY SPHERES, I 383

Fig. 4. Geometry of tmo-sphere problem showing ray that under- goes t.wo reflections.

fern obvious changes. We could carry out, the same type of analysis for a field that has undergone j reflect>ions before reaching the observer, but the analysis becomes rapidly more complicated since t.here will be, in general, 2 j - 1 unknowns involved in the form of simultaneous transcendenbal equa.tions.

It is quite difficult to describe a genera.1 hybrid ray in the sa.me sense as we did for the multiple reflected rays since it can assume any number of forms comprising creep- ing wave rays and reflected rays. However, generally, only a few contribute significant.ly to t.he scattered field in a particular direction. It. is perhaps more meaningful in describing the role of these different. types of multiple scatkered rays if we consider one ca.se in which the domi- nant mult.iple scattering mechanism is multiple reflection and anot.her in which it. is hybrid rays. In the first case we consider t.he backscat.tered field from a pair of identical metallic spheres illuminated from the broadside direct.ion, and, in the la.tter case, from the endfire direction.

BROADSIDE INCIDENCE (a = r / 2 )

Consider first the case of a pair of identical perfectly conduct.ing spheres of radius a. illuminated by a plane wave perpendicular to their common axis, a = r / 2 . For t.his discussion, we will be interested only in the back- scattered field. Some of the rays appropriate for this geometry are shown in Fig. 5. The most significant. hybrid rays for this configuration, even though t.here are four, ( X 2 1 + 2R1C-), contribute negligibly except for small spheres.

Returning to the reflected rays: the contribution due to R1 is already given in (17) wit.h 1/11 = 0. For Ra, since TJE = 1/22 = n/4 with SE = d - a@, from (18),

Fig. 5. Some rays which contribute to bachcattered field of two spheres illuminated from broadside direct.ion.

where p = d /a and the upper and lower signs refer, re- spect,ively, to horizontal a,nd vertical polarization (y = 0, n/2). This result. was also obtained by Bonkowski, et aE. [all using a lengthy tensor formulat,ion. The expressions for the fields which have undergone three, four, five, and more reflections have also been derived 1121 but. need not be written down since t.he expressions become lengthy and the recipe for obtaining them has a.lready been given.

As one would expect, the cont,ribution of the reflected ray decreases with its order. In Fig. 6, the modulus of the multiply reflected rays R? t.hrough R7 (normalized to R1 as in (19)), is shown for several values of the ratio d/a . The general behavior is perha.ps more vividly illust.rat,ed by the spot. pict.ures adjacent. to each curve. These photos were obtained by photographica.lly recording the light int.ensit,y (square of modulus) reflect.ed by tn:o polished silvered spheres. The two bright spots, common t,o a.11 the diagrams, denote the specular returns R1 from t,he front surface of each sphere. The remaining spots RZ,R3, - - -, (when they can be seen) are identified by counting inward from the two R1 spots. Each picture was obt.aiced by il- luminating the pa,ir of spheres shown in the t,op photo by a point source of ordinary light sit.ua.t.ed close to the axis of the ca.mera and recording the reflected intensit.y on film when the studio lights were extinguished. Due to the limited exposure, only those spots of intensity greater than -30 dB with respect to R1 can be seen. From t>his figure, we observe that when the spheres are separated by as little as one diameter (d/a. = 4.0), t.he magnit.udes of succeeding higher order reflected rays differ nearly by an

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, MAY 1971

I .o

v) n IO’

LL

a w c 0 W

I O - 5

I 2 3 4 5 6 7

NUMBER OF REFLECTIONS (a)

Fig. 6. Comparison of relat.ive amplitudes of multiply reflected backscattered rays for various spacings a t broadside incidence. (a) Obtained by computation. (b) 0bt.ained by photographic method (see text).

order of ma.gnitude; this is hardly the case when the spheres are in contact. It is worth remembering that the modulus of the multiple-reflected rays is a function of the ratio d/a, not, the spacing d.

Using only the rays R1, C- and Rz through Re, xve com- pute the normalized RCSs of pairs of identical metallic spheres for the tn-o principa.1 polarizations for ka = 2.00, 4.19, 6.246, and 10.00. This size range was chosen so that the results could be compared with those of the exact solution for the purpose of determining where solutions obtained by the two a.pproa.ches LLoverlap,” and also for comparison with some experimental results given in [%I. These results are presented in Fig. 7.

In the interest of comparing the two solutions as accu- rately as possible, Fig. 7 (and ones to follow) mere drawn by a computer-controlled digital plotter. A large number of data. points for each curve mas fed to the plotting program and intermediate points were calculated by a. piecewise cubic interpolation scheme. Curves comparing the two solutions were plotted at the same time on the

same grid-the dashed curves always representing the ray optical solution and the solid curves, the exact solution unless otherwise stated.

With the exception of the case of horizontal polariza- tion at ka = 2.0, the agreement is surprisingly good. From these figures, we a.lso see that as d;a becomes large (and hence the coupling small), the normalized cross section settles down to 4u,l/Ta3 as expected, ua being t.he RCS of a single sphere.

A further example is given in Fig. 8, involving the RCS of a pair of spheres in conta.ct as they both grow in size for vertical polarization. Again, the same set. of rays k considered as was used for computation in the previous example. The creeping wave influence for this example is apparent for ka 2 10 and can be identified with the local maxima in this range since me know the creeping wave C- to add in phase with R1 (for the single sphere) at, ka M 2naj ( 2 + T) , where n is a positive integer, and with a period in La of roughly %r/ ( 2 + a). Beyond this point, the RCS is almost. completely dominated by the geometric

BRUNING AND LO: NOLTIPLE SCATTERING OF WAVES BY SPHERES, I 385

k o i k b i 2 0

kd

t

ko = kb = 4.19

8 1 : : : : : : . : : : : : : : : : , b.m 10.w w m Le.m 22.00 26.w n m a.w ss.00

kd

E! : : : : : : : : : : : : : : : : , 1o.m 1u.m 1e.m n.m 26.00 90.01 3.m 38.00 u2.m kd

Lo = kb = 10.0

e l : : : : : . : : : : : : : : : : , 10.00 n.w a6.m ao.m 9u.m kd 9.m u2.m g6.m 5o.m

8 %.m t kd

?I........::::::::, 10.w n.w a6.w 3o.m %.m 38.m uz.m v6.m 5o.m kd

Fig. 7. RCS of t.wo equal perfect.ly conducting spheres at broadside incidence as spacing is varied for ka = 2.0, 4.19, 6.246, and 10.0, comparing ray-optical solution (- - -) with exact solution (-).

386 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, MAY 1971

0 - ? -

0

0 m -

0 '9-

bE 0 x - 0 x -

I , 38.00 42.00 46.00 50.00 54.00 58.00 M O O 66.00 70.00

kd

Fig. 9. (30s-polarized RCS from pair of equal perfectly conducting spheres ka = 20.0 as spacing is varied using rays F&, R,, and Rs. T and R indicate polarizations of incident and received E vectors, respectively.

optics components alone. The normalized RCS of the pair will not., however, settle down to some consta.nt value for hrge ka as it. does for the single sphere. This is because the rat.io d/a is constant, and as a result, the normalized re- turn is made up of components which are constant in magnitude; only the relative phases change with ka. Norma,lly, we associate ray met,hods with problems in which characteristic dimensions are much larger tha.n a wavelength. Here we find excellent agreement with the exact solution for a. pair of spheres in contact: even when the radii a.re as sma.ll as i/4.

There is a.not.her codguration which, because of its practical application and simplicit.y, wa.rrants mention. In this case, a.n identical pa.ir of metallic spheres is illumi- nated from t.he broadside direction with a. plane wave

whose polarization vector makes an angle of 45" wit.h the common axis of the two spheres (y = r / 4 ) . Due to symmetry, only the even components, R ~ , R ~ , s - - , R ~ ~ , con- tribute to the cross-polarized radar return. A sample com- putation for a large pa.ir of spheres (ka = 20.0) is shown in Fig. 9 where the effect. of some of the higher order rays is readily seen. The simplicity of the analysis and con- figumtion makes this an interesting method for cross- polarized RCS calibration. This is discussed more gen- erally in [25 ] with further results.

EKDFIRE INCIDENCE (a = 0) In the case of endfire incidence, as shown in Fig. 10,

t.he sphere 3 may lie wholly in the geomet,ric shadow of A if b 5 a ; hence, the only purely geometric return is the

BRUKIKG .4ND LO: MULTIPLE SCATTERING OF WAVES BY SPHERES, I 387

specular reflection from sphere A . Some rays to consider for t.his geometry when b = a a,re RI, C-, C-RPC-, CIzlC-, G-R3-YC-, C R s C - , C-C+C-, etc., where Rj" means that of j mult,iple reflections, one is a normal reflection. Strictly speaking, geometric optics dictates t,ha,t the shad- owed sphere will never see directly t.he incident, field which we know it. should eventually at very large spacing. As a result, the backscattered field of two identical spheres should eventually approach the single sphere d u e times the array factor 1 + exp (2 i kd ) .

The backscattered field due to t.he first t.wo dominant hybrid rays for arbit,rary a and b is

r d l --OQ RI A B 1.0 C.

mm C.R,C- C-R: C-

Fig. 10. Most significant rays for backscattering a t endfire in- cidence and close spacing.

.exp i2ka[1 + ((p - T)' - l)L!2 1 ilr + CSC-1 ( p - T)] - - - 2LUoac~c-~ ( p - T ) 4

(21)

where r = b / a and p = d/a. (Do4/a) is t.he product. of four surface diffraction coefficients and a. is the a.ttenu- ation coefficient associated with the normal component of the field t.hat creeps around the sphere [E].

Using the rays RI, C-, C-Rl"C-, and CAIC-, shown schematically in Fig. 10, we c0mput.e the normalized RCS of two identical meta,llic spheres for ka = 7.41, 11.048, and 20.0. These results are shown in Fig. 11 toget.her with the exact solution for compa.rison. In t.he absence of cou- pling, the normalized RCS will oscillate between 0 and 4a,/m2 as kd is varied; this 1att.er value is indicated by the dashed line on the ordinate for each case. The agree- ment gets better (for small to moderate d / a ) with in- creasing ka as we would expect, a.nd is best for kn = 20.0. The results even for ka = 7.41, however, are not, very satisfactory. The discrepancy can be a,ttribut.ed, at least in part, to the inaccuracy in the canonical creeping wa.ve problem near a. sha.dow boundary.

Let, LIS consider computing the normdized RCS of two identical metallic spheres in contact at. endfire incidence as they both grow in size using only the rays R1 a.nd C-EZPC-. I t may be recalled that the creeping wave in- fluence on the backscat.tering from a single sphere is nea.rly a.bsent at values of ka greater than about 10-15 due to its large attenuation. Placing an ident.ica1 sphere directly

k d

i ha = k b = 11.048

4.1420

kd

ha: h b . 2 0 . 0

kd

Fig. 11. RCS of t.wo equal perfectly conducting spheres at end- f i e incidence as spacing is varied for ka = 7.41, 11.048, 20.0, comparing rayoptical solut.ion (- - -) and exact solut,ion (-).

16.0 c IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, MAY 1971

ka

(b)

Fig. 12. RCS of two equal perfectly conducting spheres a t endfire incidence in contact as sphere size increases. (a) Exact. solution. (b) Ray-opt.ical solution.

behind a previously isolat.ed sphere drast.ically alters the situation, since the hybrid wave C R P C - overwhelms any single sphere creeping wave C-, and its a.ssociated creeping pat.h length is only a total of one sixth of the circumference of the first sphere, implying that the oscillations about. RI

' will persist for a much larger value of ka than for the single sphere. In Fig. 12 (a) the normalized RCS of a pair of identical spheres in contact is shown, computed by t,he mult.ipole expansion approach as ka covers the range 0 to 24. It. is clearly seen that. there is considerable oscil- lation about R1 even for ka as large as 24. It becomes somewhat costly to carry out comput,at.ions much beyond this value using the modal approach. Therefore, we must resort to ray optics. Fig. 12(b) shows t.he rayoptical solu- tion to the same problem plotted logarithmically in ka to ka = 150. If we compa.re these t.m-0 curves, we find excellent agreement in t.he locat,ion of the peaks and nulls after ka M 10 and also in the amplitude after ka M 16. The period of oscillation can be very simply determined. Know- ing that these oscillat,ions are caused only by the inter- ference of R1 and CRI1VC-, the period P , in ka, of the

oscillation can be shown from (20) to be simply a/(a/6 + ~) M 1.39, with peaks at ka = nP and nulls at ka = (2n - 1) P / 2 where n is a positive integer.3

It is interesting to note from the geometric optics point of view that eventually the front sphere can "hide" t.he back sphere at sufficiently high frequency. In fact,, the front sphere may even hide a sphere that. is larger t.han itself. Since the attenuation of the hybrid rays C_Rlh'C- is proportional to exp (- 2aoa csc-' ,.t) , ka must, be larger in order to hide a bigger sphere ( b > a ) than to hide a smaller one ( b < a ) .

COWLUSION AND DISCUSSION

The problem of electromagnet.ic scattering by t.wo spheres has been solved t,hrough two approaches: multi- pole expansion and ray optics. Numerical results show that the former solution is useful even for spheres as large as 10X in radius whereas the 1at.ter is useful for spheres as

curve for a single sphere which is 2 ~ j ( 2 + T) = 1.23. Note this is longer than the period of oscillations in the RCS

BRUNING AND LO: MUUTIPLE SCATTERING OF W.4VES BY SPHERES, I 389

small a.s X/4 in radius in some cases. The former solut.ion is very general and applicable to spheres of any mate- rial, but t,he latter is so far confined t.o conducting spheres. The ray-optical solution agrees very closely with that, of t,he multipole expansion for rehtively small spheres in the case of broadside incidence, but for much larger spheres in t.he endfire incidence case. Even for broadside incidence, the accuracy of the ray-opt,ical solution of small spheres may be good for one polarization while poor for anot.her polarization. This simply illust<rates t.hat care must be exercised in using the ray-opt,ica,l approach. Despihe this, it often gives us much physical insight into the problem.

Extension of the solut.ion to scattering of acoustic waves by two spheres is straightforward. Interested readers are referred to [l2]. It) is perhaps of interest to point out tha.t. in the case of endfire incidence, t,he solution converges considerably slower than that for the case of E M waves due to stronger creeping wave coupling. It, is also possible to extend these solut.ions to t.hree or more collinear spheres. This, as well as many other t,heoretica,l results, are discussed in [25 ] where an unusually good a,greement m<t.h t.he experiments mill be seen.

APPENDIX

TRANSLATIONAL ADDITION THEOREM FOR VECTOR SPHERICAL WAVE FUNCTIONS

In order to express vector spherica.1 wave functions about a displaced origin 0' in terms of wave functions about, another origin 0, we employ the addit.ion theorem:

m

Mmn(J'' = [A,,mnMm,(l) + Bm,mnNmp(l)] v=(l.m)

m Nmn(i)' = [Am,mnN,,(l) + Bm"mnMmp(l)]. (22)

v=(l.m)

This applies to a t.ranslat,ion along t.he z axis a distance d a5 in Fig. 1. When translating from 0' to 0, A,,"" and B,,"" are preceded by the fact,ors ( - l)m+y and ( - l)m+y+lJ respect.ively. For translat.ion in any other direction, the theorem is somewhat. more complicated a,nd mag be found elsewhere [E]-[14]. The wave funct,ions are defined in ( 5 ) and (6), and t.he Oranshtion coefficients are given by

The summation over p is finite covering t,he range I n - v I,/ n - v I + 2,--.,(n + v), and includes 1 + max { v , n ] terms. The preceding c0efficient.s are fur- t.her complica.ted by the presence of the coefficients a(nt,n, -m,v,p) which are defined by t.he linearization expansion

Pnm(~)P.-m(z) = a(m,n,-nz ,v ,p)P,(x) . (24)

These lat,t.er coefficients may be ident.ified with a product of two 3-j symbols [13], [14], [22] which are associ- ated with the coupling of two angular momentum eigen- vectors:

P

!(v - m ) ! 1'2 a(nz,n,-m,v,p) = ( 2 p + 1) (n - n7.)!(v + nt) ! 1

The factor

is the Wigner 3-j symbol of which t.here are several definitions, all involving summations of multit.udes of factorials. As a result, straight.fon.ard calculat*ion using (25) is very inefficient. I n sea,rch of a better represents.- tion, inspection of the form of the coefficients A,,"" and B,,mn in (23) reveals that a recursion rela,t,ion for the a ( .) in which only the index p cycles would be highly desirable especially for machine computation. Just such a relation exists and is given by [12]

a P 4 a d - (ap--2 + aP-1 - 4m2)a,,-2 + = 0 (26)

where

a, a.(m,n,-nz.,v,p), p = n + v,n + v - 2,.**,1n - v I and

[ ( n + v + - P T P 2 - (n - v)21. (27) ap = 4p2 - 1

We need not, be concerned with the question of st.ability of t.his recursion relation since all quantit.ies a,re rat.iona1 numbers.

The recursion relation (26) is most conveniently em-. ployed in t.he backward direction since we can find simple st.arting values for t.he coefficients a t p = n + v a.nd p = n + v - 2. By matching coefficient.s in highest. powers of the argument in (24), we find

(2n + 2v - 3) (2n - 1) (2v - 1) (n + v) an+,-? =

390 IEEE TRANS-XX‘IONS ON ANTENNAS AND PROPAGILTION, MAY 1971

[3] N. R.. Zitron and S. N. Karp, “Higher-order approximations in multiple scattering: I-two-dimensional scalar case, and II- three dimensional scalar cases,” J . Math. Phws., vol. 2, 1961.

where(2p- l ) ! ! = (29- 1) (2q-3 ) . - . 3 -1 ; ( -1 ) ! != 1. Note t.hat in (26) every new coefficient makes use of all previously calculat.ed quantities and only requires two additional evaluations of the quantit.y ap. The sta.rting values (28) a.re, of course, not calculated directly from (28), since the recurrent. form of the factorials may be ut,ilized by generating these in the proper sequence (with regard to n, v, and m) starting from n = v = 1, nt = 0, where t.he coeEcients (28) are 2/3 and 1/3, respectively. We see that nowhere ha.ve we had to calculate a single 3-j coefficient..

Special F o r m

When 7n = 0, ( 2 6 ) becomes a two-term recursion for- mula which leads to a closed expression for a(O,n,O,v,p) [lZ]. The properties of the associated Legendre funct,ions allow us to obtain a closed expression for m = 1 also, but no higher:

a n , - l,V,P)

- _ - 2 p + 1 n(n + 1) + Y ( V + 1) - p ( p + 1) 2v(v + 1 ) n + v + p + l

( - n + ; + p ) ( n - ; + p ) ( n + ; - p )

- n + v + p n - - v + p n + v - p

This expression is particularly useful as it corresponds to the case of endfire incidence.

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magnetiques par les spheres,” Ann. Phys. (Paris), 1960, pp. 265-320.

[9 1

pp. 394-406. 0. A. Germogenova, “The scat,tering of a plane electromag- netic wave by two spheres,” Izv. Akad. Nauk. SSR. Sw. Geofiz., 1963, no. 4, pp. 64H53. D. J. Angelakos and K. Kumagai, “High-frequency smktering by multiple spheres,” IEEE Trans. Antennas Propagat., vol. AP-13, Jan. 1964, pp. 105-109. 0. Lilleswet.er, “Scattering of microwaves by adjacent water droplets in air,” in Proc. F o ~ l d Con!. Radio Meterology, Sept. 1964. DD. 192-193.

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Sci., vol. 2, 1967, no. 12, pp. 1481-1495. C. && and Y . T. Lo, “Scattering by two spheres,” Radio

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[lo] V. Taersky, “On multiple scattering of waves,” J . Res. Nut . Bur. Stand., vol. 640, 1960, no. 6, pp. 715-730.

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[12] J. Bruning and Y. T. Lo, “Multiple scattering by spheres,” Antenna Lab., Univ. Illinois, Urbana, Tech. Rep. 69-5, 1969.

[13] S. Stein, “Addition t,heorems for spherical wave functions,” Quart. A p p l . Math., vol. 19, 1961, no. I, pp. 15-24.

[I41 0. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math., vol. 20, 1962,

[l5] E. Levlne and G. 0. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J . Colloid Interface

[16] J. A. Stratton, Electro??zq?%tic Theory. New York: McGraw- Hill, 1941.

[17] B. R. Levy and J. B. Keller, “Diffraction by a smooth ob- ject,” Comnzwn. Pure A p p l . Maih.., vol. 12, 1959, no. 1, pp.

[IS] W. Franz, “Uber die Greenschen funktionen des zylinders

1191 T. B. A. Senior and R. F. Goodrich, “Scattenng by a sphere,” und der kugel,” 2. Naturforsch., vol. 9, 1954, pp. 70.S716.

[20] J. H. Bruning and Y. T. Lo, “Electromagnetx scattering by Proc. Inst. Elec. Eng., vol. 3, 1964, pp. 907-916.

two spheres,” PTOC. ZEEE (Letters), vol. 56, Jan. 1968, pp. 119-120.

[21] R. R. Bonkoaski, C. R. Lubitz, and C. E. Schensted, “Studies in radar crosseection VI:-cross-sections of corner reflectors and other multiple scatterers a t microwave frequencies,” Univ. Xichigan, Ann Arbor, Tech. Rep. UMM-106, 1953.

[22] A. R. Edmonds, Angular Momendurn in Quantum Mechanics. Princeton, N. J.: Princeton Univ. Press, 1957.

[23] A. L. Aden and 3.1. Kerker, “Scattering of elect.romagnetic waves by two concentric spheres,” J . Appl. Phys., vol. 22,

[24] S. Levine and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres, when outer shell has a variable refractive index,” in Eledromagnetic Scaiiwing, M. Kerker,

[25] J. H. Bruning and Y. T. Lo, “Multiple scattering of EM Ed. Oxford, England: Pergamon, 1963.

waves byspheres,part 11-numerical and experimental results,” this issue, pp. 391-400.

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159-209.

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